Order type

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In mathematics, especially in set theory, two ordered sets X and Y are said to have the same order type if they are order isomorphic, that is, if there exists a bijection (each element pairs with exactly one in the other set) ${\displaystyle f\colon X\to Y}$ such that both f and its inverse are monotonic (preserving orders of elements).

In the special case when X is totally ordered, monotonicity of f already implies monotonicity of its inverse.

One and the same set may be equipped with different orders. Since order-equivalence is an equivalence relation, it partitions the class of all ordered sets into equivalence classes.